English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The weighted adjacency matrix $A_{f}(G)$ of a simple graph $G=(V,E)$ is the$|V|\times|V|$ matrix whose $ij$-entry equals $f(d_{i},d_j)$, where $f(x,y)$ isa symmetric function such that $f(d_i,d_j)>0$ if $ij\in E$ and $f(d_i,d_j)=0$if $ij\notin E$ and $d_i$ is the degree of the vertex $i$. In this paper, wedetermine the unique graph having the largest spectral radius of $A_{f}(G)$among all the bicyclic graphs under the assumption that $f(x,y)$ is increasingand convex in $x$ and $f(x_1,y_1)\geq f(x_2,y_2)$ when $|x_1-y_1|>|x_2-y_2|$and $x_1+y_1=x_2+y_2$. Moreover, we determine the unique graph having thesecond largest spectral radius of $A_{f}(G)$ among all the bicyclic graphs when$f(x,y)=x+y$, $(x+y)^2$ or $x^2+y^2$, which corresponds to the well-known firstZagreb index, first hyper-Zagreb index, and forgotten index, respectively. Inaddition, we also characterize the bicyclic graphs with the first two largestspectral radii of $A_{f}(G)$ when $f(x,y)=\frac{1}{2}(x/y+y/x)$, correspondingto the extended index.

Extremal spectral radius of weighted adjacency matrices of bicyclic graphs